I have been playing around with Mathematica’s Mathieu functions as I am interested in the their limit when q tends to infinity. In particular I have been plotting several asymptotic forms, the most clearly presented of which I have found to be those outlined in the work of Frenkel and Portugal http://www.lncc.br/~portugal/Mathieu.pdf
In particular I would like to point out equation (42) which I have included a comparison of
Note that the qualitative features of the Mathieu equation are butchered. Namely the solution plotted has 4 nodes, rather than 6. This happens right around q=10^9 for some reason. At a lower value of q=10^8 you can see everything works perfectly.
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